High-dimensional sample correlation matrices are a crucial class of random matrices in multivariate statistical analysis. The central limit theorem (CLT) provides a theoretical foundation for statistical inference. In this paper, assuming that the data dimension increases proportionally with the sample size, we derive the limiting spectral distribution of the matrix $\widehat{\mathbf{R}}_n\mathbf{M}$ and establish the CLTs for the linear spectral statistics (LSS) of $\widehat{\mathbf{R}}_n\mathbf{M}$ in two structures: linear independent component structure and elliptical structure. In contrast to existing literature, our proposed spectral properties do not require $\mathbf{M}$ to be an identity matrix. Moreover, we also derive the joint limiting distribution of LSSs of $\widehat{\mathbf{R}}_n \mathbf{M}_1,\ldots,\widehat{\mathbf{R}}_n \mathbf{M}_K$. As an illustration, an application is given for the CLT.

翻译：高维样本相关矩阵是多变量统计分析中一类重要的随机矩阵。中心极限定理为统计推断提供了理论基础。本文假设数据维度与样本量成比例增长，推导了矩阵$\widehat{\mathbf{R}}_n\mathbf{M}$的极限谱分布，并在两种结构——线性独立分量结构与椭圆结构——下建立了$\widehat{\mathbf{R}}_n\mathbf{M}$线性谱统计量的中心极限定理。与现有文献相比，本文提出的谱性质不要求$\mathbf{M}$为单位矩阵。此外，我们还推导了$\widehat{\mathbf{R}}_n \mathbf{M}_1,\ldots,\widehat{\mathbf{R}}_n \mathbf{M}_K$的线性谱统计量的联合极限分布。作为示例，给出了中心极限定理的一个应用。